Question

If $f$ is Lebesgue integrable on $[0,2]$ and $\int_E fdx=0$ for all measurable set E such that $m(E)=\pi/2$. Prove or disprove that $f=0$ a.e.

score 765 · Answer 1 · 2026-04-15 19:45:04

765

Views

If $f$ is Lebesgue integrable on $[0,2]$ and $\int_E fdx=0$ for all measurable set E such that $m(E)=\pi/2$. Prove or disprove that $f=0$ a.e.

Published on 15 Apr 2026 - 19:45

score 579 · Answer 2 · 2026-04-15 19:49:57

579

Views

prove $\exists$ a continuous function g s.t. $|g(x)-f(x)|<\epsilon$ for all x in a subset E of (a,b) and $\mu[(a,b)-E]<\delta$

Published on 15 Apr 2026 - 19:49

score 72 · Answer 3 · 2026-04-15 19:08:20

72

Views

Is there any example of a sequence of functions such that $\int f_n\to0$ and $\int f_n^2\nrightarrow0$?

Published on 15 Apr 2026 - 19:08

score 3.2k · Answer 4 · 2026-04-15 07:35:46

3.2k

Views

Category Theory and Lebesgue Integration.

Published on 15 Apr 2026 - 7:35

score 229 · Answer 5 · 2026-04-13 00:16:19

229

Views

antiderivative of lebesgue integrable function

Published on 13 Apr 2026 - 0:16

score 336 · Answer 6 · 2026-04-13 14:07:27

336

Views

Hölder's inequality. Understanding proof?!

Published on 13 Apr 2026 - 14:07

score 168 · Answer 7 · 2026-04-15 19:24:01

168

Views

Show $\sqrt{1 + (\int_{\Omega} h d\mu)^2} \leq \int_{\Omega} \sqrt{1+h^2} d\mu$

Published on 15 Apr 2026 - 19:24

score 100 · Answer 8 · 2026-04-13 16:52:16

100

Views

$|f(z)|^{2}\leq \frac{1}{\pi r^{2}}\iint_{D(z,r)} |f(\theta)|^{2}dm(\theta)$ for $f \in H(\Omega)$

Published on 13 Apr 2026 - 16:52

score 48 · Answer 9 · 2026-03-28 23:58:32

48

Views

$\|f+g'\|_{L^2}=\|f'-g\|_{L^2}=0\Rightarrow f=g=0$ a.e?

Published on 28 Mar 2026 - 23:58

score 181 · Answer 10 · 2026-04-12 23:39:37

181

Views

Integration of $\sin(\frac{1}{x})$

Published on 12 Apr 2026 - 23:39

score 246 · Answer 11 · 2026-04-14 12:13:01

246

Views

Is $fg\in L^1$ when $f,g\in L^1$?

Published on 14 Apr 2026 - 12:13

score 347 · Answer 12 · 2026-04-09 12:35:10

347

Views

Transformation formula for multidimensional integral

Published on 09 Apr 2026 - 12:35

score 2.1k · Answer 13 · 2026-04-14 13:39:33

2.1k

Views

Prove that function is not Lebesgue integrable

Published on 14 Apr 2026 - 13:39

score 3k · Answer 14 · 2026-04-14 03:20:36

3k

Views

Why is Lebesgue-Stieltjes a generalization of Riemann-Stieltjes? Moreover, is there an example where Lebesgue-Stieltjes is useful

Published on 14 Apr 2026 - 3:20

score 55 · Answer 15 · 2026-04-15 19:08:42

55

Views

$\forall a>0$, prove that $\int_{0}^{a} x^{-1/2} \operatorname e^{2x}\operatorname d\!x\le \frac{3}{2}a^{1/6}\operatorname e^{2a}$

Published on 15 Apr 2026 - 19:08

List Question

If $f$ is Lebesgue integrable on $[0,2]$ and $\int_E fdx=0$ for all measurable set E such that $m(E)=\pi/2$. Prove or disprove that $f=0$ a.e.

prove $\exists$ a continuous function g s.t. $|g(x)-f(x)|<\epsilon$ for all x in a subset E of (a,b) and $\mu[(a,b)-E]<\delta$

Is there any example of a sequence of functions such that $\int f_n\to0$ and $\int f_n^2\nrightarrow0$?

Category Theory and Lebesgue Integration.

antiderivative of lebesgue integrable function

Hölder's inequality. Understanding proof?!

Show $\sqrt{1 + (\int_{\Omega} h d\mu)^2} \leq \int_{\Omega} \sqrt{1+h^2} d\mu$

$|f(z)|^{2}\leq \frac{1}{\pi r^{2}}\iint_{D(z,r)} |f(\theta)|^{2}dm(\theta)$ for $f \in H(\Omega)$

$\|f+g'\|_{L^2}=\|f'-g\|_{L^2}=0\Rightarrow f=g=0$ a.e?

Integration of $\sin(\frac{1}{x})$

Is $fg\in L^1$ when $f,g\in L^1$?

Transformation formula for multidimensional integral

Prove that function is not Lebesgue integrable

Why is Lebesgue-Stieltjes a generalization of Riemann-Stieltjes? Moreover, is there an example where Lebesgue-Stieltjes is useful

$\forall a>0$, prove that $\int_{0}^{a} x^{-1/2} \operatorname e^{2x}\operatorname d\!x\le \frac{3}{2}a^{1/6}\operatorname e^{2a}$

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